Question

A sample of 16 items provides a sample standard deviation of 9.5. Test the following hypotheses using = .05. H0 : 2 50 Ha : 2 > 50 Calculate the value of the test statistic (to 2 decimals).

Answer 1

We conclude that the population standard deviation is greater than 50.

Explanation:

We are given that a sample of 16 items provides a sample standard deviation of 9.5.

Let
`\sigma^(2)` = population standard deviation

So, Null Hypothesis,
`H_0` :
`\sigma^(2) \leq` 50 {means that the population standard deviation is less than or equal to 50}

Alternate Hypothesis,
`H_A` :
`\sigma^(2)` > 50 {means that the population standard deviation is greater than 50}

The test statistics that will be used here is One-sample chi-square test statistics;

T.S. =
`((n-1) * s^(2) )/(\sigma^(2) )` ~
`\chi^(2)__n_-_1`

where, s = sample standard deviation = 9.5

n = sample of items = 16

So, the test statistics =
`((16-1) * 9.5^(2) )/(50 )` ~
`\chi^(2)__1_5`

= 27.08

The value of chi-square test statistics is 27.08.

Now, at 5% level of significance the chi-square table gives a critical value of 25.00 at 15 degrees of freedom for the right-tailed test.

Since the value of our test statistics is more than the critical value of chi as 27.08 > 25.00, so we have sufficient evidence to reject our null hypothesis as it will fall in the rejection region.

Therefore, we conclude that the population standard deviation is greater than 50.